Viscosity, Entropy and the Viscosity to Entropy Density Ratio; How Perfect Is a Nucleonic Fluid?
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Viscosity, entropy and the viscosity to entropy density ratio; how perfect is a nucleonic fluid? Aram Z. Mekjian Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08502, TRIUMF Laboratory, 4004 Westbrook Ave. , Vancouver, B.C. CA V6T 2A3 Abstract The viscosity of hadronic matter is studied using a classical evaluation of the scattering angle and a quantum mechanical discussion based on phase shifts from a potential. Semi classical limits of the quantum theory are presented. A hard sphere and an attractive square well potential step are each considered as well as the combined effects of both. The lowest classical value of the viscosity for an attractive potential is shown to be a hard sphere limit. The high wave number-short wavelength limits of the quantum result have scaling laws associated with it for both the viscosity and entropy. These scaling laws are similar to the Fraunhoher diffraction increase for the hard sphere geometric cross section. Specific examples for nuclear collisions are given. The importance of the nuclear tensor force and hard core is mentioned. The viscosityη , entropy density s andη / s ratio are calculated for a gas of dilute neutrons in the unitary limit of large scattering length. Away from the unitary limit, the ratio of the interaction radius or the scattering length to the interparticle spacing introduces a variable y besides the fugacity z . The isothermal compressibility is shown to impose important constraints. The results forη / s are compared to the AdS/CFT string theory minimum of (1/ 4π )h / kB to see how close a nucleonic gas is to being a perfect fluid. Theη / s ~1 h / kB for a neutron gas in its unitary limit. Theη / s ~ 3 h / kB treating the nuclear scattering as billiard ball collisions. The minimumη / s for a neutron gas occurs in regions of negative isothermal compressibility and high fugacity where higher virial terms are important. In a neutron-proton system higher virial terms are associated with a liquid-gas phase transition and critical opalescent phenomena. A connection between the nuclear flow tensor and viscosity is developed using a Fokker-Planck equation and a relaxation time description. The type of flow- laminar, vortex, turbulent- is investigated. PACS 24.10.Pa, 21.65.Mn, 66.20.-d I. Introduction Understanding the viscosity of matter is of interest in many regimes of energy and in several different areas of physics. These areas include atomic systems, nuclear matter, neutron star physics, low energy to relativistic energy heavy ion collisions and at the extreme end string theory. For example viscosity plays a role in collective flow phenomena in medium and relativistic heavy ion collisions (RHIC). Specifically, viscosity inhibits or resists flow in such collisions. Ref. [1,2] are early theoretical studies of viscosity and flow. Some recent experimental results for RHIC physics can be found in Ref. [3] and some overviews are in Ref. [4,5]. A microscopic kinetic theory description 1 of viscosity involves the transport of momentum across an area and involves particle interactions through mean free path arguments. If interactions are strong, the shear viscosity is small. The shear viscosity is inversely proportional to the scattering cross section. Substances with low kinematic viscosity and with high Reynolds number do not flow smoothly as in laminar flow, but rather form eddies when flowing around obstacles. The Reynolds number also depends on the flow velocity and the eddy behavior is easily seen by putting a paddle in water in a boat moving at various speeds. Surprisingly, in relativistic heavy ion collisions, the quarks and gluons act as a strongly coupled liquid [6,7,8] with low viscosity rather than a nearly ideal gas of asymptotically free particles with high viscosity. Low viscosity fluids which interact strongly are called nearly perfect fluids. String theory has put a small lower limit on the ratio of shear viscosityη over entropy density s given byη / s ≥ h /(4πkB ) [9] with kB the Boltzmann constant. The string theory result has generated considerable interest in questions concerning strongly correlated systems and viscosity. The focus of the present paper is on low and moderate energy systems of interacting nucleons. A system of nucleons has parallel properties such as correlated behavior, bound states, phase transitions and critical point behavior. Nearly perfect fluids with very low viscosity appear also in cold atoms. Properties of interacting quantum degenerate Fermi gases were first observed in atomic systems [10- 12]. Atoms are cooled in a laser trap to the point where quantum statistics and an associated thermal wavelength play an important role. Such systems can be tuned by using a magnetic field and properties of Feshbach resonances are used to study the strong coupling crossover from a Bose –Einstein condensate of bound pairs to a Bardeen- Cooper-Scrieffer BCS superfluid state of Cooper pairs. A remarkable aspect of strongly interacting Fermi gases is a universal behavior which occurs when the scattering length is very large compared to the interparticle spacing. In this unitary limit, properties of a heated gas are determined by the density ρ and temperatureT , independent of the details of the two body interaction. Early theoretical discussions of dilute Fermi systems at infinite scattering length can be found in Ref. [13,14]. At temperatureT = 0 , the Fermi energy E of a strongly interacting Fermi gas differs from the Fermi energy EF of a non- interacting Fermi gas by a universal factorξ with E = ξEF . Accounting for this difference in nuclear systems is referred to as the Bertsch challenge problem [15]. Initial work on this problem was done by Barker [16] and latter Heiselberg [17]. A Monte Carlo numerical study of the unitary limit of pure neutron matter is given in Ref. [18]. Analytic studies of pure neutron systems can be found in the extensive work of Bulgac and collaborators [19-21]. In these studies the dimensionless factorξ ≈ 0. 4 . Part of present paper is an extension of an earlier work [1] in which the viscosity of hadronic matter was studied using a relaxation time approximation to the Boltzmann equation and also a Fokker-Planck description. This earlier paper focused on features associated with collective flow. Specifically, the kinetic flow tensor was related to the pressure tensor and the collective velocity field. The pressure tensor, in turn, was related to the nuclear viscosity. The Kubo-Green formulae [22,23,24] relates the viscosity to time fluctuations in the pressure tensor. A calculation of the Reynolds number showed that the flow is laminar. A Fokker Planck approach has been recently used [25] to study the viscosity of the quark-gluon phase. A relaxation time approach also appears in Ref. [26,27] for trapped Fermi gas in a oscillator well near the unitary limit of large scattering 2 length. Questions related to entropy in nuclear systems were also studied early on [28] using the Sakur-Tetrode law and the Saha-equation. Recently, the behavior of entropy in the unitary limit was given [29]. The work presented in this paper gives a more refined calculation of the viscosity than given in Ref. [1] as well as further discussions of the entropy. Both classical and a quantum approaches to the viscosity are discussed using a Chapman-Enskog description [30,31]. The classical calculation involves the scattering angle while the quantum approach relates the viscosity to properties of the scattering phase shifts. In such studies a potential must be specified. The interactions used here to describe the nuclear scattering are a square well potential, a pure hard core and a combination of the two. The pure hard core potential pictures the collisions as arising from impenetrable billard balls and is used as a comparison. The discussion of viscosity from a pure hard core potential also contains new results which show a scaling law regime with increasing momentum of the colliding pair of nucleons. This scaling feature parallels results that arise when considering scattering from a hard sphere [32] which lead to a diffractive increase by a factor of two from the classical result ofπR 2 . The square well potential with a hard core approximates a nuclear potential and the role of the nuclear hard core as well as the nuclear tensor force are mentioned . A study of viscosity using a delta-shell potential can be found in Ref.[33,34]. The focus here will be on a pure one component system- such as in a gas of neutrons. One important feature of the interaction between nucleons is the very large scattering length asl and its associated unitary regime. The unitary limit of the viscosity and entropy is examined for this system. A study of Feshbach resonances and the second virial coefficient in atomic systems is given in ref. [35,36]. Viscosity to entropy considerations also appear in the damping of giant resonances in nuclear physics [37] and in atoms in laser traps [38]. II. Classical and quantum approaches to the viscosity II. A. Mean free path and relaxation time approaches First, the standard discussion of the viscosity that can be found in textbooks such as Ref. [39] relate the viscosity to the number density ρ , mass of a fluid particle m , mean ) speed v = 8k BT / mπ of a Boltzmann distribution and mean free pathlλ as 1 η = ρmv)l . (1) 3 λ 2 The mean free pathlλ = 1/(ρσ ) . The scattering cross sectionσ = πD for hard sphere scattering of particles with diameter D . For this description the viscosityη no longer depends on the number density ρ and is simply 1 1 η = mv) . (2) 3 πD2 ) Theη does not involve Planck’s constant. If the mean free path is taken aslλ = vτ , with 3 τ the mean time between collisions, theη ~ ετ .